In Alec-Stashevsky/hhjboot: Select an Optimal Block-Length to Bootstrap Dependent Data (Block Bootstrap)

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set.seed(32)

blocklength

blocklength is an R package used to automatically select the block-length parameter for a block-bootstrap. It is meant for use with dependent data such as stationary time series.

The Story

Regular bootstrap methods rely on assumptions that observations are independent and identically distributed (i.i.d.), but this assumption fails for many types of time series because we would expect the observation in the previous period to have some explanatory power over the current observation. This could occur in any time series from unemployment rates, stock prices, biological data, etc. A time series that is i.i.d. would look like white noise, since the following observation would be totally independent of the previous one (random).

To get around this problem, we can retain some of this time-dependence by breaking-up a time series into a number of blocks with length l. Instead of sampling each observation randomly (with replacement) like a regular bootstrap, we can resample these blocks at random. This way within each block the time-dependence is preserved.

The problem with the block bootstrap is the high sensitivity to the choice of block-length, or the number of blocks to break the time series into.

The goal of blocklength is to simplify and automate the process of selecting a block-length to perform a bootstrap on dependent data. blocklength has several functions that take their name from the authors who have proposed them. Currently, there are three methods available:

1. hhj() takes its name from the Hall, Horowitz, and Jing (1995) "HHJ" method to select the optimal block-length using a cross-validation algorithm which minimizes the mean squared error (MSE) incurred by the bootstrap at various block-lengths.

2. pwsd() takes its name from the Politis and White (2004) Spectral Density "PWSD" Plug-in method to automatically select the optimal block-length using spectral density estimation via "flat-top" lag windows of Politis and Romano (1995).

3. nppi() takes its name from the Lahiri, Furukawa, and Lee (2007) Nonparametric Plug-In "NPPI" method to select the optimal block-length for block bootstrap procedures. The NPPI method estimates the leading term in the first-order expansion of the theoretically optimal block length by using resampling methods to construct consistent bias and variance estimators for the block-bootstrap. Specifically, this package implements the Moving Block Bootstrap (MBB) method of Künsch (1989) and the Moving Blocks Jackknife (MBJ) of Liu and Singh (1992) as the bias and variance estimators, respectively.

Under the hood, hhj() uses the moving block bootstrap (MBB) procedure according to Künsch (1989) which resamples blocks from a set of overlapping sub-samples with a fixed block-length. However, the results of hhj() may be generalized to other block bootstrap procedures such as the stationary bootstrap of Politis and Romano (1994).

Compared to pwsd(), hhj() is more computationally intensive as it relies on iterative sub-sampling processes that optimize the MSE function over each possible block-length (or a select grid of block-lengths), while pwsd() is a simpler "plug-in" rule that uses auto-correlations, auto-covariance, and the spectral density of the series to optimize the choice of block-length. Similarly, nppi() is another "plug-in" rule, however, due to its heavy reliance on resampling, it can also be computationally intensive compared to pwsd().

For a detailed comparison, see the table below:

# Comparison table
table <- data.frame(
  rows = c(
    "**Method Type**",
    "**Computational Cost**",
    "**Primary Goal**", 
    "**Variance Estimation**",
    "**Bias Estimation**",
    "**Best for**", 
    "**Estimation Capacity**",
    "**Dependency\\***"
    ),
  NPPI = c(
    "Nonparametric resampling",
    "Medium (bootstrap resampling & jackknife)",
    "Minimize MSE of bootstrap estimator",
    "Moving Blocks Jackknife-After-Bootstrap (JAB)", 
    "Directly estimates bias from bootstrap",
    "General-purpose estimators, small sample sizes, and quantile estimation", 
    "Bootstrap bias, variance, distribution function, and quantile estimation",
    "User-defined parameters for initial block-length `l` and number of deletion blocks `m`"
    ),
  PWSD = c(
    "Spectral density estimation",
    "Low (direct ACF computation)", 
    "Estimate block length using spectral density",
    "Implicitly estimated via spectral density", 
    "Indirectly accounts for bias via ACF decay",
    "Block-length selection for circular and stationary bootstrap, time series with strong autocorrelation", 
    "Bootstrap sample mean only",
    "User-defined parameters for autocorrelation lag and implied hypothesis tests (4 total)"
    ),
  HHJ = c(
    "Subsampling-based cross-validation",
    "High (subsampling & cross-validation)", 
    "Minimize MSE via cross-validation",
    "Uses subsample-based variance estimation", 
    "Uses subsample-based bias estimation",
    "Estimating functionals with strong dependencies", 
    "Bootstrap variance and distribution function estimation", 
    "Requires user-defined parameters for `pilot_block_length` (*l\\**) and `sub_sample` size (*m*)"
    )
)

# Render table
knitr::kable(
  table,
  format = "markdown",
  col.names = c(
    "",
    "NPPI (Lahiri et al., 2007)",
    "PWSD (Politis & White, 2004)",
    "HHJ (Hall, Horowitz & Jing, 1995)"
    ),
  caption = "* All algorithms have default user-defined parameters recomended by the respective authors."
  )

Installation

You can install the released version from CRAN with:

install.packages("blocklength")

You can install the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("Alec-Stashevsky/blocklength")

Use Case

We want to select the optimal block-length to perform a block bootstrap on a simulated autoregressive AR(1) time series.

First we will generate the time series:

library(blocklength)

# Simulate AR(1) time series
series <- stats::arima.sim(model = list(order = c(1, 0, 0), ar = 0.5),
                           n = 500, rand.gen = rnorm)

# Coerce time series to data.frame (not necessary)
data <- data.frame("AR1" = series)

Now, we can find the optimal block-length to perform a block-bootstrap. We do this using the three available methods.

1. The Hall, Horowitz, and Jing (1995) "HHJ" Method

## Using the HHJ Algorithm with overlapping subsamples of width 10
hhj(series, sub_sample = 10, k = "bias/variance")

2. The Politis and White (2004) Spectral Density Estimation "PWSD" Method

# Using Politis and White (2004) Spectral Density Estimation
pwsd(data)

We can see that both methods produce similar results for a block-length of 9 or 11 depending on the type of bootstrap method used.

3. The Lahiri, Furukawa, and Lee (2007) Nonparametric Plug-In "NPPI" Method

# Using Lahiri, Furukawa, and Lee (2007) Nonparametric Plug-In 
nppi(data, m = 8) 

Acknowledgements

A big shoutout to Malina Cheeneebash for designing the blocklength hex sticker! Also to Sergio Armella and Simon P. Couch for their help and feedback!

Alec-Stashevsky/hhjboot documentation built on Feb. 19, 2025, 11:19 a.m.